March 2001 Contents
Analyzing Games of Information
Gone into are twenty questions, the counterfeit coin, ranking tennis players, and Mastermind. An algorithm for the last, code available from the author, lets you win the game with 4.3426 guesses. On the average, that is.
Simpson's Rule with Constant Weights
R. S. Pinkham
The coefficients in Simpson's Rule are 1, 2, 4, 2, 4, ... , 4, 2, 1. This is curious: why should some interior points count more than others? The author modifies the rule so that the interior points all have equal weight. Alas, the error term doubles. You can't have everything.
Verhulst's Logistic Curve
A new way of looking at the logistic differential equation, making it easier to solve. No more partial fractions!
Conceptions of Area: In Students and in History
Bronislaw Czarnocha, Ed Dubinsky, Sergio Loch, Vrunda Prabhu, and Draga Vidakovic
One way of getting at area is the method of exhaustion: chop it up into pieces that get smaller and smaller until all the area is gone. Another is the method of indivisibles: add up all the infinitesimal pieces to get the total. Students' intuitions about area do not always go along with how we teach them about it.
The Band Around a Convex Set
Tie a rope around the equator so it fits snugly. Then add 20 meters to its length, distribute it evenly around the circumference, and the rope will be a bit more than 3 meters above the surface all the way around. This is easy to show. Harder to show, but not that much harder, is that the same phenomenon occurs for any convex set.
Fallacies, Flaws, and Flimflam
A proof that is correct, but is mysterious, and a second proof that explains, but unfortunately is wrong.
A Series for ln k
1 + 1/2 - 2/3 + 1/4 + 1/5 - 2/6 + ... = ln 3. 1 + 1/2 +1/3 - 3/4 + 1/5 + 1/6 + 1/7 - 3/8 + ... = ln 4. What next? And why is it so?
A Game-like Activity for Learning Cantor's Theorem
Fun is not always to be found in mathematics, but there is something to be said for maximizing it. Here is a game-like idea for making Cantor's theorem about the inequivalence of a set and its power set comprehensible. (The diagonal method often baffles students.)
Take a 15-dimensional space (if you can find one) and slice through it with six subspaces of dimension 14. The author gives two methods for seeing how many pieces you get.
Image Reconstruction in Linear Algebra
Olympia Nicodemi and Andrzej Kedzierawski
A camera photographs a scene, blurring some of the details. It would be nice to be able to reconstruct the original. Here is a simplified version of the problem, with arrays of pixels to which linear algebra may be applied.
Linear Relations Between Powers of Terms in Arithmetic Progression
The solution to 3x + 5y = 7, 9x + 11y = 13 is (-1, 2) and the same is true for any 2-by-2 system with coefficients in arithmetic progression. If the coefficients of a 3-by-3 system are squares of numbers in arithmetic progression, the solution is always (1, -3, 3). You know what's coming next, don't you? But I bet you couldn't prove the general result without reading this Capsule.
ActivStats is a program by Paul Velleman that, as its name implies, is meant to teach statistics and to involve its users actively. I'll give away the ending of the review: it's a good program.
Problems and Solutions
In recent years there has been a relative spate of books about specific numbers. Brian Blank takes a look at six of them, variously devoted to 0, p, i, and e. As he notes, a book on 1 is overdue.